Before Facebook and Google became the megaliths of the web, the most famous online adage was, "on the internet, no one knows you're a dog". It seems the days when people were allowed to be dogs is coming to a close. The old web, a place where identity could remain separate from real life, is rapidly disappearing from the computer screen. According to Sheryl Sandberg, Facebook's chief operating officer, and Richard Allan, its director of policy in Europe, a critical mass of people only want online interactions supported by "authentic" identity. And this, say critics, will have irrevocable effects on the openness of the web.

The pursuit of authenticity is creeping into the heart of most social media models and in the current internet landscape is playing an important role in how we engage with one another and with web content. For many people, Facebook and Google products are the sum total of their web interaction, and the value in creating a platform that provides confidence that a person is who they say they are, rather someone pretending to be them, is critical to a social network's success.

Within this model, authentic identity is non-anonymous. Facebook profiles and Google IDs are tied to a person's real name and real connections, and increasingly to their activities across cyberspace. Users are familiar with logging into other services using Facebook or Google IDs, forming a single public identity that's an aggregated version of their offline past, the online present and their combined future.

Facebook also believes authenticity is linked to a person's photo stream, which is why it has just paid $1bn for the photo-sharing service Instagram. "Pictures speak a thousand words," says Allan. "Immigration officials will ask to see a photo album to see if a relationship is genuine. It's a very instinctive and powerful way to confirm authentic identity."

Not everyone agrees. "I would not call what you have on Facebook 'authentic' identity," says Christopher Poole, the 24-year-old creator of 4Chan, an online community founded in 2004. 4Chan boasts two design features antithetical to Facebook: first, its 20 million users don't register an account to participate and are therefore anonymous; second, there's no archive.

Poole, who was voted Time's most influential person of 2008 – two years before Facebook's Mark Zuckerberg was declared the magazine's Man of the Year – believes Facebook's commercial motivations shut down the online experience: "Mark and Sheryl have gone out and said that identity is authenticity, that you are online who you are offline, and to have multiple identities is lacking in integrity. I think that's nuts."

"We went from a web that was interest-driven, and then we transitioned into a web where the connections were in-person, real-life friendship relationships," adds Poole. "Individuals are multifaceted. Identity is prismatic, and communities like 4Chan exist as a holdover from the interest-driven web."

Allan believes such attitudes are naive. The millions who have gone online over the past decade want a safe place where they won't experience bad behaviour, have their identities stolen or be duped by impostors, he says: "Pretend identities don't work very well now that the web has moved from a minority sport for geeks to a mainstream occupation."

And this attitude is baked into the main players' systems: any profile on Facebook or Google that does not appear to be tied to an offline name is removed. Nicknames and pseudonyms, regardless of their longevity – and some have been in use for decades – are considered breaches of terms of service. What people do online now, and will be doing in the foreseeable future, is inherently tied to their offline selves. And this locks down what it is considered acceptable to do and who it is acceptable to meet.

Yet a social network's success need not rely upon this direct link between online and offline identity. In Japan, the three most popular social networks operate under pseudonyms at the discretion of the account holder. "[Japanese social networks are] anonymous, but we can trace past mentions by login ID or nickname," explains Yasutaka Yuno, editor-in-chief of Japan's most popular mobile technology site, K-tai Watch. "The past mentions are useful to judge credibility. In each social community, ID acts as personal name."

An online identity can be as permanent as an offline one: pseudonymous users often identify themselves in different social networks using the same account name. But because their handles aren't based on real names, they can deliberately delineate their identity accordingly, and reassert anonymity if they wish. Psychologists argue that this is valuable for the development of a sense of who one is, who one can be, and how one fits into different contexts. This kind of activity is allowed even in countries where social network account holders are required to register for a service using a national ID, as in South Korea and China; their online public identities are still fabrications. Even with this explicit link with the state, when users are aware that their activities online are traceable, identity play continues.

Andrew Lewman, executive director of the Tor Project, hopes to re-anonymise the web. "The ability to be anonymous is increasingly important because it gives people control, it lets them be creative, it lets them figure out their identity and explore what they want to do, or to research topics that aren't necessarily 'them' and may not want tied to their real name for perpetuity," he says.

The Tor browser and software obfuscates a user's web traffic so anyone watching is unable to trace who a user is or where they are coming from, by bouncing an individual's communications through at least three different places. People can still be identifiable on a service like Facebook or Google if they choose to log in, but Tor prevents these sites knowing what users were doing before, and where they go after they log out.

This is a technological solution to what Lewman feels is an elemental problem with the de-anonymisation of the web. "The ability to forget, to start over is important," he argues. "Maybe you just got divorced, maybe you just came out of rehab and you want to start over.

"As soon as you log into a Gmail account, you start getting ads for the drug rehab you want to forget. If you're in a real-name environment, such as Facebook, unless you actually physically change your name and your friends, you're thrown right back into your old life."

Although Facebook does allow users to curate what's public and private – "recasting your public persona by selecting from the data you've put onto the service," explains Allan – Lewman believes the automated systems make a total social reinvention difficult to pull off.

"If you truly wanted to be anonymous, you'd have to use a combination of 4Chan and Tor," explains Poole. Tor provides the back-end anonymity that compliments 4Chan's front-end anonymity. But this is technologically advanced: it is the major players setting the identity agenda. And so the ideological battle over online identity continues.

"Facebook is setting the expectations of what we want," says Poole. "They set the bar in terms of what kind of control their users have over their identity online. They've been moving that bar slowly but surely in a direction that they might call transparency, but what other people might call lack of choice."

Allan believes the benefits of authentic identity outweigh the costs. Facebook and other services with an assurance of security and credibility are more inclusive, and open up the web to new audiences who never would have gone online before, he says. "We're optimists. Facebook enables hundreds of millions of people to express themselves online because they didn't have or know how to use the tools they needed." Facebook, he believes, is a stepping stone to the rest of the web.

And if they are successful at promoting their particular brand of authentic identity, if you want to be a dog on the internet in the future, you'll have to have papers to prove it.